Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C215 |
| direct product | C2×#72236790 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×#127 |
| commutator | C23×C4 | C24×C4 |
| Frattini, cosocle | C24×C4 | C25 |
| socle | C24 | C2×#392 |
| prime radical | C22×#36614 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
At least 7749 conjugacy classes of subgroups; only the 7768 normal subgroups are shown.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,8,9,10)(11,12)(13,21,22,26)(14,18)(15,19)(16,20)(17,23,25,24);(3,4)(5,6)(8,10)(11,13)(12,17,18,25)(14,22)(15,24)(16,23)(19,26,20,21);(12,18)(19,20);(4,6)(12,18)(17,25)(19,20)(21,26)320 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (3,4)(5,6)(7,8,9,10)(11,12)(13,21,22,26)(14,18)(15,19)(16,20)(17,23,25,24), (3,4)(5,6)(8,10)(11,13)(12,17,18,25)(14,22)(15,24)(16,23)(19,26,20,21), (12,18)(19,20), (4,6)(12,18)(17,25)(19,20)(21,26));